Sarah never smiled in my classroom. At least not with her eyes. She spoke under her breath. Whispers of words I couldn’t usually hear. She is one of the many students who I remember because despite my best efforts, I wasn’t able to help her see herself as a mathematical thinker, or to build her curiosity and mathematical skills.
For much of my teaching career, I taught eighth grade math students who needed extra support. My co-teacher and I were tasked with ensuring the students covered all the curricular components of first year Algebra. Most of the students already hated math and believed that they were bad at it and would never improve. They had received poor grades in the past, found math to be boring, and expected much of the same from us.
How could I blame them? Many students experienced math as disembodied, dry and onerous. Math class was a state of perpetual confusion and frustration. These students were polite and kind. They didn’t say these things to me. But I could see it in the way Tina slipped away whenever possible for a drink at the water fountain, the way Joe sat with his shoulders drooping and the way Samantha took her tests, her head on her arms, silent tears slipping down her cheeks.
We started every year with a review of fraction operations. The students groaned. My heart sank. Understanding of fractions is highly predictive of success in higher level math. They were unsure if 1/20 was bigger or smaller than 2/5. I tried to spark conversation and curiosity. How big is a 20th? What if we cut something into 100 pieces? 1000 pieces? Would those pieces be smaller or bigger? We moved on. The curriculum always had to move on, dragging along with it a majority of students who didn’t understand the last unit and were now thrown into a new one. I knew that if a student didn’t understand fractions in September, they weren’t going to understand negative exponents in March (spoiler alert: fractions again!). This type of learning is a house of cards. We’d set them up.
In May, our students took a standardized test on Algebra, and we always had a bell curve. Teachers are often applauded for a bell curve. I never understood that. I didn’t want a bell curve. I wanted everyone to master the material. This felt imperative to me. Anything else twisted my stomach and felt unethical. How could I let the majority of my students face high school without basic numeracy and even lower self confidence?
What Does Mastery Mean?
To develop mastery, students must acquire foundational skills, practice using them together, and know when to apply what they have learned.
Learning is contextual. Generalizing learning to new contexts (what education researchers call transfer) requires facilitation. Many people over the years told me that mastery is unrealistic. There would always be students who didn’t earn an A. “But don’t we want them to master middle school math?” I would reply. “Do we really not believe that all of these students are capable of understanding fractions?”
My favorite rebuttals were the ones that argued not every student belonged in honors math. “If they all have A’s,” this thinking went, “how will we know who can really do next level work?”
As Sanjay Sarma points out in his book “Grasp,” the American education system has long been about “winnowing” down masses of people to those who are deemed deserving of educational investment and those who are not. Nowhere is this more clear than in mathematics education. Often as early as fourth or fifth grade students are sorted into different “tracks” for math with different paces, levels of depth, and unspoken expectations of student long term success. From elementary school through high school and college, these tracks often become “traps” that reinforce systemic inequalities in our society. We are witnesses to “goal displacement;” the sorting of students, originally meant to assist in delivering better instruction, has now become the end goal itself.
Another popular defense of the status quo was that the students needed to learn some “lesson” from the poor grade, such as how to take better notes, study more effectively or work more quickly on a test. This argument was some version of “tough love,” where we were helping students experience the consequences of their actions. Except we rarely took time to explicitly teach those skills or discuss them. How would they know how to succeed next time if we didn’t teach them? Another popular reply was that we simply didn’t have time. Yet the pace of our teaching meant students had surface-level understanding, retained concepts poorly, and frequently required reteaching.
My experience teaching math took place at academically strong independent schools with an abundance of resources. If we cannot ensure students’ mastery of quantitative literacy in this context, where and when is it possible?
Creating a system that allows all students the opportunity for mastery of fundamental mathematical concepts and confidence in their own mathematical reasoning is not easy, but we must try.
The current system forces everyone into the same timetable to reach proficiency. This ignores individual differences as well as the last several decades of cognitive science research. Our current model of math instruction misses the opportunity to elevate generations of students with the joy and power of numeracy.
Teaching is an incredibly difficult and complex job. However, there is a great deal that we do know about human learning that can help us to imagine a better system.
We can begin with the shared high expectation that all students in our classroom have the ability to achieve mastery of foundational mathematical skills and habits of mind. We can make mastery the outcome of learning and not a given amount of time in a classroom. We can adjust our classrooms and assessments so that students move through material at the pace that is best suited to them and achieve mastery of skills they can use for the rest of their lives.
Some individuals and institutions are already making these changes. For example, The Modern Classrooms Project provides professional development for teachers on mastery-based classroom instructional models and the Mastery Transcript Consortium is a network of schools reimagining the high-school transcript.
We must accelerate this shift before another generation of talented students are winnowed down and the majority cast aside.
